I wonder, if you are asking two different questions:
Why don’t you receive notifications about updated packages?
Two: Security and bugfixes
For 1. it depends which desktop environment you use, Gnome/KDE have this update notifications out of the box, for other DEs (Xfce, LXDE, etc.) you might need to enable this with the installation of synaptic or similar.
For 2. Debian stable does not ship bugfixes but Debian stable ships security fixes. I highly recommend to subscribe to Debians Security mailing list, especially for security fixes concerning browsers and other stuff.
Edit: I have enabled automatic updates and I still receive regular notifications via Gnome Software, at least once per week.